Spreads and the symmetric topos Original Research Article

We introduce a new intrinsic notion of spread for toposes and geometric morphisms, and use it to give a “topological” characterization of Lawvere distributions on a topos. In the process, we relate spreads to zero-dimensional locales, and establish two new pure/spread factorizations for geometric morphisms. Our results are then applied to the study of the symmetric topos as a generalized lower power locale. In particular, we show that the symmetric topos is part of a Kock-Zöberlein 2-monad on toposes, give a new construction of bicomma squares in which the “lower” leg is an essential geometric morphism, characterize local connectedness in terms of the symmetric topos, and relate the symmetric and bagdomain constructions via the classifier of probability distributions.